Standard +0.3 This is a straightforward application of standard summation formulas. Part (a) requires algebraic manipulation of known results for Σr and Σr³, which is routine for FP1 students. Part (b) uses the 'hence' technique of subtracting two evaluations, a standard exam pattern. While it involves multiple steps, no novel insight is required—just careful execution of familiar techniques.
6. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } r \left( 2 r ^ { 2 } - 6 \right) = \frac { 1 } { 2 } n ( n + 1 ) ( n + 3 ) ( n - 2 ) .$$
(b) Hence calculate the value of \(\sum _ { r = 10 } ^ { 50 } r \left( 2 r ^ { 2 } - 6 \right)\).
6. (a) Use the standard results for $\sum _ { r = 1 } ^ { n } r$ and for $\sum _ { r = 1 } ^ { n } r ^ { 3 }$ to show that, for all positive integers $n$,
$$\sum _ { r = 1 } ^ { n } r \left( 2 r ^ { 2 } - 6 \right) = \frac { 1 } { 2 } n ( n + 1 ) ( n + 3 ) ( n - 2 ) .$$
(b) Hence calculate the value of $\sum _ { r = 10 } ^ { 50 } r \left( 2 r ^ { 2 } - 6 \right)$.\\
\hfill \mbox{\textit{Edexcel FP1 Q6 [6]}}